Picard groups and torsion-free cancellation for orders in $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$

Abstract

We examine the question of direct-sum cancellation of finitely generated, torsion-free modules over ring orders $R$ contained in $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$. Using conditions involving the Picard group and the unit group of $R$, we give a nearly complete classification of those orders $R$ for which torsion-free cancellation holds. There is exactly one `exceptional' order to which our methods do not apply.